1 數位控制（五）

2 data hold

3 x(t) G(s) y(t) In continuous-time In discrete-time g(kT) is the system ’ s weighting sequence g(t) is the system ’ s weighting sequence x(t) G(s) y(t)x*(t) X(z)

4 pulse transfer function Transfer function continuous-time system Laplace transform Pulse transfer function discrete-time system z transform G(z) X(z) Y(z)

5 x(t) G(s) y(t) x*(t) X(z) x(t) G(s) y(t)

6 Pulse transfer function of cascade elements x(t) y(t) x*(t)u*(t) x(t) y(t) x*(t)

7 Pulse transfer function of closed-loop system R(s) G(s) C(s) E*(s) + - H(s) E(s)

8 Pulse transfer function of a digital controller

9 Closed-loop pulse transfer function r(t) Plant c(t) e(kT) + - e(t) Digital Controller Zero-order hold D/A R(s) Plant C(s) E*(s) + - E(s) G* D (s) Sampler A/D m(kT)u(t) M*(s)

10 PID Controller Proportional-Integral-Derivative (PID) Controller Most traditional and has been use successfully for over 50 years. P: action proportional to the actuating error between the input and the feedback signal. I: action proportional to the integral of the actuating error D: action proportional to the derivative of the actuating error

11 Analog PI Controller

12 Foxboro 控制器之 PIDA Block 規劃 FW Flow Set FW Flow OUT LAG

13 PID Controller Ziegler-Nichols Stability Boundary Tuning 方式，係於 如右上圖之閉迴路 下，逐漸調高 Proportional Gain ， 直至 K u 產生如右中 圖所示之震盪為止， 量測震盪週期 P u ， 再依所示公式估算 最佳化 P 與 I 值。

14 負載響應 在控制參數已調整至最佳化下，該波形之過衝量 Ω (overshoot) 以及衰減比 δ (decay ratio) 皆應趨近於 0.2 。

15 PB 值最佳化調整 比例帶 Proportional Band (PB)= 100/P. PB 主要決定負載波形之減 幅 (damping) 以及對稱性， 左圖顯示在最佳化積分時間 下不同 PB 設定下之負載波 形： 較小之 PB(P opt /1.5) 會造成衰 減比 (decay ratio) 較大，亦即 減幅太輕 (light damping) 波 形，應調高其 PB 。 太大之 PB(P opt *1.5) 會造成第 一個波峰呈現非對稱波形並 需較長時間回到穩態，應調 低其 PB 。 圖、不同比例帶 PB 之負載響應波形

16 I 值最佳化調整 積分時間 I 值主要決定 負載波形之過衝量 (overshoot) ， 左圖顯示在最佳化 PB 下不同 I 值設定下之負 載波形： 較小之 I(Iopt/1.5) 會造 成較大之過衝 (overshoot) ，應調高其 I 值。 較大之 I(Iopt*1.5) 會造 成沒有過衝，亦即 undershoot ，應調低其 I 值。 圖、不同積分時間 I 之負載響應波形

17 Pulse transfer function of a digital PID controller R(s) Plant C(s) E*(s) + - E(s) G* D (s) M*(s)

18 Discretize the m(t) (positional form)

19 Transient response of PID Controller R(s) C(s) E*(s) + - E(s) G* D (s) M*(s) R(z) C(z) + -

20 Realization of digital controllers and filters

21 Direct programming

22 Standard programming

23 Series programming

24 Parallel programming

25 Infinite vs Finite